Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for first-order arithmetic sentences, so I'm not sure what it means to deduce a $\Pi^m_n$ or $\Sigma^m_n$ sentence using the $\omega$-rule for $m>0$ (this may be in the linked paper I don't have access to) - but in fact there's a coarse calculation which will apply to any reasonable interpretation I can think of:

Every version of the $\omega$-rule I can think of is $\Pi^1_1$ - roughly, "the $\omega$-consequences of $T$" will always be $\Pi^1_1$ relative to $T$. If (for example) we start with the true $\Pi^1_1$ theory of arithmetic, we won't even get all the true $\Sigma^1_1$ sentences since the true $\Sigma^1_1$ theory of arithmetic isn't itself $\Pi^1_1$ (more broadly, the projective hierarchy doesn't collapse).

Now your notion of specialness gives us only two tools for "climbing up" the syntactic hierarchy: deduction and complementation. In terms of Turing degree this means that we're not going to escape the $\omega$th hyperjump of $\emptyset$, which is a tiny subclass of $\Delta^1_2$.

(This is contra a silly claim I made originally - the point is that "$\Pi^1_1$ in $\Sigma^1_1$" is much weaker than "$\Pi^1_2$." The sense in which applying the $\omega$-rule "adds a $\Pi^1_1$" seems to follow the former pattern if set up in a natural way. That said, I see no sense at all in which we can get outside the analytical hierarchy.)

Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for first-order arithmetic sentences, so I'm not sure what it means to deduce a $\Pi^m_n$ or $\Sigma^m_n$ sentence using the $\omega$-rule for $m>0$ (this may be in the linked paper I don't have access to) - but in fact there's a coarse calculation which will apply to any reasonable interpretation I can think of:

Every version of the $\omega$-rule I can think of is $\Pi^1_1$ - roughly, "the $\omega$-consequences of $T$" will always be $\Pi^1_1$ relative to $T$. If (for example) we start with the true $\Pi^1_1$ theory of arithmetic, we won't even get all the true $\Sigma^1_1$ sentences since the true $\Sigma^1_1$ theory of arithmetic isn't itself $\Pi^1_1$ (more broadly, the projective hierarchy doesn't collapse).

Now your notion of specialness gives us only two tools for "climbing up" the syntactic hierarchy: deduction and complementation. In terms of Turing degree this means that we're not going to escape the $\omega$th hyperjump of $\emptyset$, which is a tiny subclass of $\Delta^1_2$.

(This is contra a silly claim I made originally - the point is that "$\Pi^1_1$ in $\Sigma^1_1$" is much weaker than "$\Pi^1_2$," or more concretely that $\mathcal{O}^\mathcal{O}$ is not $\Pi^1_2$ complete. The sense in which applying the $\omega$-rule "adds a $\Pi^1_1$" seems to follow the former pattern if set up in a natural way. That said, I see no sense at all in which we can get outside the analytical hierarchy.)

Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for first-order arithmetic sentences, so I'm not sure what it means to deduce a $\Pi^m_n$ or $\Sigma^m_n$ sentence using the $\omega$-rule for $m>0$ (this may be in the linked paper I don't have access to) - but in fact there's a coarse calculation which will apply to any reasonable interpretation I can think of:

Every version of the $\omega$-rule I can think of is $\Pi^1_1$ - roughly, "the $\omega$-consequences of $T$" will always be $\Pi^1_1$ relative to $T$. If (for example) we start with the true $\Pi^1_1$ theory of arithmetic, we won't even get all the true $\Sigma^1_1$ sentences since the true $\Sigma^1_1$ theory of arithmetic isn't itself $\Pi^1_1$ (more broadly, the projective hierarchy doesn't collapse).

Now your notion of specialness gives us only two tools for "climbing up" the syntactic hierarchy: deduction and complementation. Deduction is only going to "add a $\Pi^1_1$" here, so we get that the special classes are exactly the $\Sigma^m_n$s and $\Pi^m_n$s for $m\in\{0,1\}$ and $n\in\omega$; that is, special = analytical.

Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for first-order arithmetic sentences, so I'm not sure what it means to deduce a $\Pi^m_n$ or $\Sigma^m_n$ sentence using the $\omega$-rule for $m>0$ (this may be in the linked paper I don't have access to) - but in fact there's a coarse calculation which will apply to any reasonable interpretation I can think of:

Every version of the $\omega$-rule I can think of is $\Pi^1_1$ - roughly, "the $\omega$-consequences of $T$" will always be $\Pi^1_1$ relative to $T$. If (for example) we start with the true $\Pi^1_1$ theory of arithmetic, we won't even get all the true $\Sigma^1_1$ sentences since the true $\Sigma^1_1$ theory of arithmetic isn't itself $\Pi^1_1$ (more broadly, the projective hierarchy doesn't collapse).

Now your notion of specialness gives us only two tools for "climbing up" the syntactic hierarchy: deduction and complementation. In terms of Turing degree this means that we're not going to escape the $\omega$th hyperjump of $\emptyset$, which is a tiny subclass of $\Delta^1_2$.

(This is contra a silly claim I made originally - the point is that "$\Pi^1_1$ in $\Sigma^1_1$" is much weaker than "$\Pi^1_2$." The sense in which applying the $\omega$-rule "adds a $\Pi^1_1$" seems to follow the former pattern if set up in a natural way. That said, I see no sense at all in which we can get outside the analytical hierarchy.)